Learning Proofs

Date: 12/19/95 at 18:0:5
From: Anonymous
Subject: Books on Proofs

Dear Dr. Math,

Hi, I'm a fourth grader interested in learning how to do proofs.
Can you recommend some good books on the different techniques
used and how they are applied?  It doesn't have to be only
about geometry - I like algebra too.

Thanks,

Erin

Date: 6/17/96 at 9:46:17
From: Doctor Mike
Subject: Re: Books on Proofs

Erin,  If we had been able to respond immediately, you could have 
put SOLVE IT! by James F. Fixx (1978, Doubleday) right at the top 
of your holiday gifts wish list.  Also good, but in a totally
different way, is MATH PROJECTS FOR YOUNG SCIENTISTS by
David A. Thomas (1988, Franklin Watts).
  
What a proof is depends on whether you are talking about 
math, science, law, politics, etc.  Many good books on 
proof assume experience from high school or beyond.  
I recommend getting as much math-related experience 
as you can.  Solving so-called word problems is best. 
  
Problem solving is really the first stage of developing
an understanding of mathematical proof.  That's why the
Fixx book is so good. When you finally figure out a
problem, you understand it and can explain it, and you
just know by common sense that you are right. The
PROJECTS book is more open-ended. Check in your library
for titles concerning puzzles, brainteasers, paradoxes
or mathematical recreations.  Books you find by Martin
Gardner or Raymond Smullyan also are worth a look.
  
The Thomas Y. Crowell Co. has a YOUNG MATH BOOKS 
series of several dozen short books "on rather sophisticated
subjects, introduced on an easy-to-understand enjoyable
level for the youngest of mathematicians."  I have seen
several, and really like BASE FIVE by David Adler and
LESS THAN NOTHING IS REALLY SOMETHING by Robert Froman.
YES-NO; STOP-GO by Judith Gersting and Joseph Kuczkowski
explains truth tables.  This looks like a fine series.
  
When you are ready to move on from practice in problem
solving to studying the actual techniques of valid
arguments and clearly presenting conclusive evidence,
the paperback HOW TO READ AND DO PROOFS by Daniel Solow
(1982, John Wiley) is excellent.  It is well-written,
but might be rough going without help from a teacher or
a friend who already understands many of these ideas.

Another math doctor suggested HOW TO SOLVE IT by George
Polya (1957, Doubleday).  

-Doctor Mike,  The Math Forum